On the connectivity of the escaping set for complex exponential Misiurewicz parameters

Abstract:

Let $E_{\lambda }(z)=\lambda \textrm {exp}(z), \lambda \in \mathbb C$, be the complex exponential family. For all functions in the family there is a unique asymptotic value at $0$ (and no critical values). For a fixed $\lambda$, the set of points in $\mathbb C$ with orbit tending to infinity is called the escaping set. We prove that the escaping set of $E_{\lambda }$ with $\lambda$ Misiurewicz (that is, a parameter for which the orbit of the singular value is strictly preperiodic) is a connected set.